Theorem 2.6 ([1]}) 
Let the graph \(G=(V,E)\)  be of maximum degree \({\Delta }>0\) . Suppose that \(G\)  can be embedded in a surface
of Euler genus  \(g\ge 0\) . If \({\Delta }\ge d(g)+2\) , then
\(\nonumber {\rho }({\bf A}_G)\le \sqrt{8({\Delta }-d(g))}+d(g),\) 
